FIRST AND SECOND ONLY. Mathematical formulation with constraints only.

A garden furniture company has 4 products: garden tables, garden chairs, garden benches, and parasols. A garden table takes 8 hours to produce with the material cost of 60. A garden chair takes 6 hours to make with the material cost of 20. A garden bench takes 4 hours to make with the material cost of 25. A parasol takes 2 hours to make with the material cost of 10. Each week, there are 800 hours available with a weekly budget of 3000 for material costs. To make maintain diversity of the products, the company will produce at least 5 units for each product. A garden table typically comes with 4 chairs, whereas a bench may also have 4 chairs. Thus, the number of chairs should be at least 4 times the total number of tables and benches. Some customers may also wish to buy a parasol if they buy a table. Thus, the number of parasols should not exceed the number of tables. The profits per unit are 260, 140, 60, and 50 for tables, benches, parasols, and chairs, respectively. The main objective is to organize the production portfolio so as to maximize the profit of the company on a weekly basis. a) Formulate this problem as an LP with the objective and all the appropriate constraints. b) Solve the LP and find the maximum profit and discuss your result. c) The company seems to do well, so it decides to hire more people (thus more available hours) with the same material cost budget. If the number of available hours is now 1000, will it affect the production portfolio? Should the company recruit more people? Discuss and explain. d) Another strategy is to increase the budget for raw materials to 5000 per week with the available hours unchanged. Will this strategy increase the profit? Discuss your findings. e) The company later realizes that the garden benches do not sell well, so it may consider not to produce any benches at all. Will this potential change increase or decrease the profit (assuming that the other conditions remain the same)? Discuss and explain briefly